{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:P2KB4GCNAONTYAJ6AXSNPODLNX","short_pith_number":"pith:P2KB4GCN","schema_version":"1.0","canonical_sha256":"7e941e184d039b3c013e05e4d7b86b6dd58dffe4ba6635d7c520a2517af680a7","source":{"kind":"arxiv","id":"1809.00824","version":2},"attestation_state":"computed","paper":{"title":"A Liouville Theorem on the PDE $\\det(f_{i\\bar j})=1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"An-Min Li, Li Sheng","submitted_at":"2018-09-04T07:56:20Z","abstract_excerpt":"Let $f$ be a smooth plurisubharmonic function which solves $$ \\det(f_{i\\bar j})=1\\;\\;\\;\\;\\;\\;\\mbox{in }\\Omega\\subset \\mathbb C^n.$$ Suppose that the metric $\\omega_{f}=\\sqrt{-1}f_{i\\bar j}dz_{i}\\wedge d\\bar z_{j}$ is complete and $f$ satisfies the growth condition $$ C^{-1}(1+|z|^2)\\leq f\\leq C(1+ |z|^2),\\;\\;\\;\\; as\\;\\;\\; |z|\\to \\infty. $$ for some $C>0,$ then $f$ is quadratic."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.00824","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-09-04T07:56:20Z","cross_cats_sorted":[],"title_canon_sha256":"a0e477b81698ab7b8dbd26a766e9f2cdea2b8ebd19887bbb1f554ddfdca8c501","abstract_canon_sha256":"8436baddb81638352b75797e1d341a3a89de89f91b06fc69d3b29deb3ebf37e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:28.299278Z","signature_b64":"4auqcaXpvaVVs5w1/K9w8Wi6uYyfp4C4jhrwBuqkSeZ9YucRkYw+Jr/YsKfvnKiyX9KXoZTqNxTKkB6vE726AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e941e184d039b3c013e05e4d7b86b6dd58dffe4ba6635d7c520a2517af680a7","last_reissued_at":"2026-05-18T00:06:28.298702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:28.298702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Liouville Theorem on the PDE $\\det(f_{i\\bar j})=1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"An-Min Li, Li Sheng","submitted_at":"2018-09-04T07:56:20Z","abstract_excerpt":"Let $f$ be a smooth plurisubharmonic function which solves $$ \\det(f_{i\\bar j})=1\\;\\;\\;\\;\\;\\;\\mbox{in }\\Omega\\subset \\mathbb C^n.$$ Suppose that the metric $\\omega_{f}=\\sqrt{-1}f_{i\\bar j}dz_{i}\\wedge d\\bar z_{j}$ is complete and $f$ satisfies the growth condition $$ C^{-1}(1+|z|^2)\\leq f\\leq C(1+ |z|^2),\\;\\;\\;\\; as\\;\\;\\; |z|\\to \\infty. $$ for some $C>0,$ then $f$ is quadratic."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00824","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.00824","created_at":"2026-05-18T00:06:28.298779+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.00824v2","created_at":"2026-05-18T00:06:28.298779+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00824","created_at":"2026-05-18T00:06:28.298779+00:00"},{"alias_kind":"pith_short_12","alias_value":"P2KB4GCNAONT","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"P2KB4GCNAONTYAJ6","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"P2KB4GCN","created_at":"2026-05-18T12:32:43.782077+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX","json":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX.json","graph_json":"https://pith.science/api/pith-number/P2KB4GCNAONTYAJ6AXSNPODLNX/graph.json","events_json":"https://pith.science/api/pith-number/P2KB4GCNAONTYAJ6AXSNPODLNX/events.json","paper":"https://pith.science/paper/P2KB4GCN"},"agent_actions":{"view_html":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX","download_json":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX.json","view_paper":"https://pith.science/paper/P2KB4GCN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.00824&json=true","fetch_graph":"https://pith.science/api/pith-number/P2KB4GCNAONTYAJ6AXSNPODLNX/graph.json","fetch_events":"https://pith.science/api/pith-number/P2KB4GCNAONTYAJ6AXSNPODLNX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX/action/storage_attestation","attest_author":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX/action/author_attestation","sign_citation":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX/action/citation_signature","submit_replication":"https://pith.science/pith/P2KB4GCNAONTYAJ6AXSNPODLNX/action/replication_record"}},"created_at":"2026-05-18T00:06:28.298779+00:00","updated_at":"2026-05-18T00:06:28.298779+00:00"}